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## G = C14.562+ 1+4order 448 = 26·7

### 56th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C14.562+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22⋊D28 — C14.562+ 1+4
 Lower central C7 — C2×C14 — C14.562+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C14.562+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=a7b2d >

Subgroups: 1436 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, Q8×C14, C23×D7, Dic74D4, C22⋊D28, D14.5D4, C4⋊D28, C4⋊C4⋊D7, C2×D14⋊C4, C287D4, C28.23D4, C7×C22⋊Q8, C14.562+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, Q82D7, C23×D7, D46D14, C2×Q82D7, D48D14, C14.562+ 1+4

Smallest permutation representation of C14.562+ 1+4
On 112 points
Generators in S112
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 61 26 84)(2 62 27 71)(3 63 28 72)(4 64 15 73)(5 65 16 74)(6 66 17 75)(7 67 18 76)(8 68 19 77)(9 69 20 78)(10 70 21 79)(11 57 22 80)(12 58 23 81)(13 59 24 82)(14 60 25 83)(29 95 55 102)(30 96 56 103)(31 97 43 104)(32 98 44 105)(33 85 45 106)(34 86 46 107)(35 87 47 108)(36 88 48 109)(37 89 49 110)(38 90 50 111)(39 91 51 112)(40 92 52 99)(41 93 53 100)(42 94 54 101)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 80)(58 81)(59 82)(60 83)(61 84)(62 71)(63 72)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 45 26 33)(2 44 27 32)(3 43 28 31)(4 56 15 30)(5 55 16 29)(6 54 17 42)(7 53 18 41)(8 52 19 40)(9 51 20 39)(10 50 21 38)(11 49 22 37)(12 48 23 36)(13 47 24 35)(14 46 25 34)(57 110 80 89)(58 109 81 88)(59 108 82 87)(60 107 83 86)(61 106 84 85)(62 105 71 98)(63 104 72 97)(64 103 73 96)(65 102 74 95)(66 101 75 94)(67 100 76 93)(68 99 77 92)(69 112 78 91)(70 111 79 90)
(1 40 8 33)(2 41 9 34)(3 42 10 35)(4 29 11 36)(5 30 12 37)(6 31 13 38)(7 32 14 39)(15 55 22 48)(16 56 23 49)(17 43 24 50)(18 44 25 51)(19 45 26 52)(20 46 27 53)(21 47 28 54)(57 95 64 88)(58 96 65 89)(59 97 66 90)(60 98 67 91)(61 85 68 92)(62 86 69 93)(63 87 70 94)(71 107 78 100)(72 108 79 101)(73 109 80 102)(74 110 81 103)(75 111 82 104)(76 112 83 105)(77 99 84 106)```

`G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,61,26,84)(2,62,27,71)(3,63,28,72)(4,64,15,73)(5,65,16,74)(6,66,17,75)(7,67,18,76)(8,68,19,77)(9,69,20,78)(10,70,21,79)(11,57,22,80)(12,58,23,81)(13,59,24,82)(14,60,25,83)(29,95,55,102)(30,96,56,103)(31,97,43,104)(32,98,44,105)(33,85,45,106)(34,86,46,107)(35,87,47,108)(36,88,48,109)(37,89,49,110)(38,90,50,111)(39,91,51,112)(40,92,52,99)(41,93,53,100)(42,94,54,101), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,80)(58,81)(59,82)(60,83)(61,84)(62,71)(63,72)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,45,26,33)(2,44,27,32)(3,43,28,31)(4,56,15,30)(5,55,16,29)(6,54,17,42)(7,53,18,41)(8,52,19,40)(9,51,20,39)(10,50,21,38)(11,49,22,37)(12,48,23,36)(13,47,24,35)(14,46,25,34)(57,110,80,89)(58,109,81,88)(59,108,82,87)(60,107,83,86)(61,106,84,85)(62,105,71,98)(63,104,72,97)(64,103,73,96)(65,102,74,95)(66,101,75,94)(67,100,76,93)(68,99,77,92)(69,112,78,91)(70,111,79,90), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,107,78,100)(72,108,79,101)(73,109,80,102)(74,110,81,103)(75,111,82,104)(76,112,83,105)(77,99,84,106)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,61,26,84)(2,62,27,71)(3,63,28,72)(4,64,15,73)(5,65,16,74)(6,66,17,75)(7,67,18,76)(8,68,19,77)(9,69,20,78)(10,70,21,79)(11,57,22,80)(12,58,23,81)(13,59,24,82)(14,60,25,83)(29,95,55,102)(30,96,56,103)(31,97,43,104)(32,98,44,105)(33,85,45,106)(34,86,46,107)(35,87,47,108)(36,88,48,109)(37,89,49,110)(38,90,50,111)(39,91,51,112)(40,92,52,99)(41,93,53,100)(42,94,54,101), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,80)(58,81)(59,82)(60,83)(61,84)(62,71)(63,72)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,45,26,33)(2,44,27,32)(3,43,28,31)(4,56,15,30)(5,55,16,29)(6,54,17,42)(7,53,18,41)(8,52,19,40)(9,51,20,39)(10,50,21,38)(11,49,22,37)(12,48,23,36)(13,47,24,35)(14,46,25,34)(57,110,80,89)(58,109,81,88)(59,108,82,87)(60,107,83,86)(61,106,84,85)(62,105,71,98)(63,104,72,97)(64,103,73,96)(65,102,74,95)(66,101,75,94)(67,100,76,93)(68,99,77,92)(69,112,78,91)(70,111,79,90), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,107,78,100)(72,108,79,101)(73,109,80,102)(74,110,81,103)(75,111,82,104)(76,112,83,105)(77,99,84,106) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,61,26,84),(2,62,27,71),(3,63,28,72),(4,64,15,73),(5,65,16,74),(6,66,17,75),(7,67,18,76),(8,68,19,77),(9,69,20,78),(10,70,21,79),(11,57,22,80),(12,58,23,81),(13,59,24,82),(14,60,25,83),(29,95,55,102),(30,96,56,103),(31,97,43,104),(32,98,44,105),(33,85,45,106),(34,86,46,107),(35,87,47,108),(36,88,48,109),(37,89,49,110),(38,90,50,111),(39,91,51,112),(40,92,52,99),(41,93,53,100),(42,94,54,101)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,80),(58,81),(59,82),(60,83),(61,84),(62,71),(63,72),(64,73),(65,74),(66,75),(67,76),(68,77),(69,78),(70,79),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(1,45,26,33),(2,44,27,32),(3,43,28,31),(4,56,15,30),(5,55,16,29),(6,54,17,42),(7,53,18,41),(8,52,19,40),(9,51,20,39),(10,50,21,38),(11,49,22,37),(12,48,23,36),(13,47,24,35),(14,46,25,34),(57,110,80,89),(58,109,81,88),(59,108,82,87),(60,107,83,86),(61,106,84,85),(62,105,71,98),(63,104,72,97),(64,103,73,96),(65,102,74,95),(66,101,75,94),(67,100,76,93),(68,99,77,92),(69,112,78,91),(70,111,79,90)], [(1,40,8,33),(2,41,9,34),(3,42,10,35),(4,29,11,36),(5,30,12,37),(6,31,13,38),(7,32,14,39),(15,55,22,48),(16,56,23,49),(17,43,24,50),(18,44,25,51),(19,45,26,52),(20,46,27,53),(21,47,28,54),(57,95,64,88),(58,96,65,89),(59,97,66,90),(60,98,67,91),(61,85,68,92),(62,86,69,93),(63,87,70,94),(71,107,78,100),(72,108,79,101),(73,109,80,102),(74,110,81,103),(75,111,82,104),(76,112,83,105),(77,99,84,106)]])`

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28X order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 28 28 28 28 4 ··· 4 14 14 14 14 28 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D14 2+ 1+4 Q8⋊2D7 D4⋊6D14 D4⋊8D14 kernel C14.562+ 1+4 Dic7⋊4D4 C22⋊D28 D14.5D4 C4⋊D28 C4⋊C4⋊D7 C2×D14⋊C4 C28⋊7D4 C28.23D4 C7×C22⋊Q8 C22⋊Q8 C2×C14 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C14 C22 C2 C2 # reps 1 2 2 2 2 2 1 1 2 1 3 4 6 9 3 3 2 6 6 6

Matrix representation of C14.562+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 10 10 0 0 0 0 19 22 0 0 0 0 0 0 0 10 0 0 0 0 26 3
,
 28 2 0 0 0 0 28 1 0 0 0 0 0 0 28 0 3 12 0 0 0 28 5 12 0 0 1 28 1 0 0 0 2 22 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 28 1 28 0 0 0 27 7 0 28
,
 17 0 0 0 0 0 0 17 0 0 0 0 0 0 9 14 0 0 0 0 19 20 0 0 0 0 0 0 1 9 0 0 0 0 3 28
,
 12 0 0 0 0 0 12 17 0 0 0 0 0 0 9 14 0 0 0 0 15 20 0 0 0 0 0 0 24 14 0 0 0 0 19 5

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,0,26,0,0,0,0,10,3],[28,28,0,0,0,0,2,1,0,0,0,0,0,0,28,0,1,2,0,0,0,28,28,22,0,0,3,5,1,0,0,0,12,12,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,28,27,0,0,0,1,1,7,0,0,0,0,28,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,9,19,0,0,0,0,14,20,0,0,0,0,0,0,1,3,0,0,0,0,9,28],[12,12,0,0,0,0,0,17,0,0,0,0,0,0,9,15,0,0,0,0,14,20,0,0,0,0,0,0,24,19,0,0,0,0,14,5] >;`

C14.562+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{56}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.56ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1097);`
`// by ID`

`G=gap.SmallGroup(448,1097);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,184,675,297,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^7*b^-1,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;`
`// generators/relations`

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